3 answers

From what I see, the Task API returns task data (unless you input parts=None), while claim only returns one thing. That may make claim a faster API to call (unless it is a wrapper API).

There could be the possibility that both use different security setups. Claim may only work if you are in the Team a task is assigned to, while Assign Task to me can claim the task if you have the credentials to reassign any task.

Claim may be designed specifically to prevent people from stepping over one another.

If you assign or claim a task does not make much difference, is just a wording thing. We usually use claim for a assignment the user does himself/herself, while assign to is can also done by another person/group.

However there are differences in authorization configuration:

Claim a task (PUT) /task/{taskID}?action={string} is restricted by the task action ACTION_CLAIM